6/04/2009 @ 10:20AM

The Olympics of Voting

In Olympic events the best is unambiguous. The race goes to the swiftest, the match to the strongest.

Not so in the voting for the Olympic host city. On Oct. 2 the International Olympic Committee will meet in Copenhagen to choose the site for the 2016 Summer Games. Chicago is among the contenders, along with Madrid, Tokyo and Rio de Janeiro. Some would even say that Chicago is the favorite, assisted by Barack Obama’s victory in November. So suppose Chicago is the best option for 2016. Does that mean it will necessarily be chosen? Don’t bet on it.

In selecting a host city, the IOC, acting like a papal conclave, takes a series of votes until a candidate receives a majority. Each of the 100-plus IOC members gets one vote, and after every round the city with the fewest votes is eliminated. In the competition for the 2012 Games, Moscow expired in the first round, New York in the second. Madrid was the top vote-getter in Round 2, but it got the ax in Round 3. London edged out Paris, 54–50, in the final vote.

Let’s suppose Chicago really is the favorite for 2016: It’s the highest-ranked city on average, and it beats all the others in head-to-head votes. Voters can rank the four candidates in 24 different ways, but say they sort themselves into five equal-size factions with the preferences in SET A (as in the chart below).

As one can easily confirm, Chicago is highest-rated city on average, and it beats all the others in paired votes. Everybody likes Chicago a lot–but nobody likes it most of all, and under the IOC’s rule Chicago is the first to go. Tokyo is the eventual winner.

But the IOC’s rule can also work to Chicago’s advantage, even if it’s not deserving. Imagine that the five blocs instead rank the cities as in SET B.

Now Chicago wins the 2016 Games–Round 1 eliminates Tokyo, Round 2 Rio and Round 3 Madrid–even though, given that set of preferences, Chicago is a poor choice. Three blocs rank it near the bottom; two think it’s worst of all. It’s not the highest-rated city on average–Madrid is. And the only city Chicago beats in a paired vote is Madrid.

But maybe all this really shows is that the IOC should scrap its current rule and come up with another. Wouldn’t it be better to choose the highest-ranked city on average? Well, not so fast. Let’s imagine the IOC’s voting blocs have the orderings in SET C.

Tokyo’s average ranking is the highest, so it prevails under the new rule. But suppose the IOC had made Doha a finalist instead of Rio, and imagine the voting blocs rank the four cities this way, as in SET D.

The Olympics of Voting

SET A

Bloc 1

Bloc 2

Bloc 3

Bloc 4

Bloc 5

Madrid

Tokyo

Rio

Rio

Tokyo

Chicago

Chicago

Chicago

Chicago

Chicago

Tokyo

Madrid

Tokyo

Madrid

Madrid

Rio

Rio

Madrid

Tokyo

Rio

SET B

Bloc 1

Bloc 2

Bloc 3

Bloc 4

Bloc 5

Madrid

Rio

Madrid

Chicago

Chicago

Rio

Tokyo

Tokyo

Madrid

Madrid

Tokyo

Chicago

Rio

Tokyo

Rio

Chicago

Madrid

Chicago

Rio

Tokyo

SET C

Bloc 1

Bloc 2

Bloc 3

Bloc 4

Bloc 5

Madrid

Chicago

Tokyo

Madrid

Chicago

Tokyo

Tokyo

Chicago

Tokyo

Tokyo

Chicago

Madrid

Madrid

Chicago

Madrid

Rio

Rio

Rio

Rio

Rio

SET D

Bloc 1

Bloc 2

Bloc 3

Bloc 4

Bloc 5

Madrid

Chicago

Tokyo

Madrid

Chicago

Tokyo

Doha

Chicago

Tokyo

Doha

Chicago

Tokyo

Madrid

Chicago

Tokyo

Doha

Madrid

Doha

Doha

Madrid

Nothing has changed in the rankings except Rio’s replacement by Doha: the relative ratings of Chicago, Madrid and Tokyo are exactly the same. But everything has changed in the outcome: Chicago now has a higher average than Tokyo, and it wins the 2016 Games. Thank you, Doha!

Okay, so maybe the IOC picked the wrong rule again. What if it awards the Games to the city that beats all the others in head-to-head votes? Sorry. Paired votes can “cycle,” yielding, for example, Chicago over Rio, Rio over Madrid, Madrid over Tokyo and Tokyo over Chicago. Nobody wins!

For centuries philosophers, mathematicians, political scientists and economists have searched for the best method of voting. Fifty-eight years ago the economist Kenneth Arrow (later a Nobel laureate) decided to see whether any voting rule could avoid the problems we’ve illustrated. Fix them all at once, he found, and you get–a dictatorship. One voter calls the shots every time. Arrow’s “impossibility theorem” demonstrates that no system of voting always gives the “right” result.

In the Olympics of voting, there are no gold medalists.

John Mark Hansen is a political scientist and Allen R. Sanderson is an economist, both at the University of Chicago.